How can i create a piece of ground on a cylinder like the one on the picture without having random things happening on the sides?
The following two methods might help to fix your problem:
Method 1
Scale down a cylinder to be the size of your current mesh. After that go in Edit Mode, select all of your mesh by pressing A, and then Subdivide. Set the number of cuts to three. Once that is done, add a Subdivision Surface Modifier and set it to 2 or 3 (Don't forget to add Loop Cuts on the top and bottom of the mesh by hitting Ctrl+R and then dragging. Otherwise you will get a very round mesh) If you are in cycles you can set the feature set to experimental and enable adaptive subdivision in the Subdivision Surface modifier and then in the material properties go to Settings and then set bump only to displacement and bump to save render times.
Then give that cylinder the same material you gave to your current mesh.
Method 2
If you want your current mesh to have the material, then your problem is that your UV's are stretched. Make sure you have given seams on the right places. To make sure there is no stretch, go to the UV Editing Workspace and in Edit Mode select your entire mesh. Then on the side panel on the left, go to Overlays and press display stretch (The colours any other than almost dark blue like cyan or green mean there is stretch).

If there is any stretching, you will have to first go to Object Mode and then press Ctrl+A and click Apply Scale. Come back to Edit Mode, then select your entire mesh and press U &gt; Unwrap. If the stretching is still visible you would have to add more seams at the right place and then hit U &gt; Unwrap again.